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\begin{document}
\title{Math 127 Homework}
\author{Mary Radcliffe}
\date{Due 31 January 2019}
\maketitle

Complete the following problems. Fully justify each response. You need only turn in those problems marked with a (*).

\begin{enumerate}

\item (*) Let $a, b, c$ be integers. Prove that if $a$ divides $b$ and $b$ divides $c$, then $a$ also divides $c$. 

\item (*) Let $n, m$ be integers. Prove that $n+m$ is odd if and only if one of $n, m$ is even and the other is odd.

\item (*) Find all real solutions $x$ to the equation $\sqrt{x+1}+\sqrt{x-3}=4$. Prove that your answer is correct.

\item Follow the method of Example 9 to prove that if $p$ is a positive prime integer, then $\sqrt{p}$ is irrational.

\item Let $x$ be a rational number and $y$ an irrational number. Prove that $x+y$ is irrational using the method of contradiction.

\item (*) Complete the proof of Example 12: that is, show that if $a, b, c$ are all odd integers, and $k$ and $\ell$ are both integers, and $ak^2+bk\ell+c\ell^2=0$, then $k$ and $\ell$ are both even.

\item Using Example 6 as a model, show that if $n$ is a positive integer, then $n$ is divisible by 9 if and only if the sum of the base 10 digits of $n$ is divisible by 9.

\item (*) Use the method of proof by contradiction to prove that there are infinitely many prime numbers. You may wish to use the result from Example 11 to help you.

\end{enumerate}
\end{document}